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| Mirrors > Home > MPE Home > Th. List > crnggrpd | Structured version Visualization version GIF version | ||
| Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.) | 
| Ref | Expression | 
|---|---|
| crngringd.1 | ⊢ (𝜑 → 𝑅 ∈ CRing) | 
| Ref | Expression | 
|---|---|
| crnggrpd | ⊢ (𝜑 → 𝑅 ∈ Grp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | crngringd.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
| 2 | 1 | crngringd 20244 | . 2 ⊢ (𝜑 → 𝑅 ∈ Ring) | 
| 3 | 2 | ringgrpd 20240 | 1 ⊢ (𝜑 → 𝑅 ∈ Grp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 Grpcgrp 18952 CRingccrg 20232 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-ring 20233 df-cring 20234 | 
| This theorem is referenced by: fermltlchr 21545 psdmul 22171 psd1 22172 psdpw 22175 ply1chr 22311 ply1fermltlchr 22317 elrgspnsubrunlem1 33252 elrgspnsubrunlem2 33253 erlbr2d 33269 rlocaddval 33273 rloccring 33275 rloc0g 33276 rlocf1 33278 fracerl 33309 evl1deg1 33602 evl1deg2 33603 evl1deg3 33604 ply1dg1rt 33605 irngss 33738 irredminply 33758 algextdeglem4 33762 algextdeglem5 33763 aks6d1c1p3 42112 aks6d1c2lem4 42129 aks6d1c6lem2 42173 aks5lem2 42189 evlsbagval 42581 selvvvval 42600 evlselv 42602 selvadd 42603 prjcrv0 42648 | 
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